Data Structures and Algorithms in Java[1].pdf - Fulvio Frisone

where a and b are integers and a ≤ b. Summations arise in data structure and
algorithm analysis because the running times of loops naturally give rise to
summations.
Using a summation, we can rewrite the formula of Proposition 4.3 as
.
Likewise, we can write a polynomial f(n) of degree d with coefficients a 0 , …, a d
as
.
Thus, the summation notation gives us a shorthand way of expressing sums of
increasing terms that have a regular structure.
4.1.7 The Exponential Function
Another function used in the analysis of algorithms is the exponential function,
f(n) = b n ,
where b is a positive constant, called the base, and the argument n is the exponent.
That is, function f(n) assigns to the input argument n the value obtained by
multiplying the base b by itself n times. In algorithm analysis, the most common
base for the exponential function is b = 2. For instance, if we have a loop that starts
by performing one operation and then doubles the number of operations performed
with each iteration, then the number of operations performed in the nth iteration is
2 n . In addition, an integer word containing n bits can represent all the nonnegative
integers less than 2 n . Thus, the exponential function with base 2 is quite common.
The exponential function will also be referred to as exponent function.
We sometimes have other exponents besides n, however; hence, it is useful for us to
know a few handy rules for working with exponents. In particular, the following
exponent rules are quite helpful.
Proposition 4.4 (Exponent Rules): Given positive integers a,b,and
c,we have
1. (b a ) c = b ac
2. b a b c = b a+c
3. b a /b c = b a − c .
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